Unraveling the role of dipolar versus Dzyaloshinskii-Moriya interactions in stabilizing compact magnetic skyrmions

We present a theoretical study of compact magnetic skyrmions in ferromagnetic films with perpendicular magnetic anisotropy that accounts for the full stray field energy in the thin film and low interfacial Dzyaloshinskii-Moriya interaction (DMI) regime. In this regime, the skyrmion profile is close to a Belavin-Polyakov profile, which yields analytical expressions for the equilibrium skyrmion radius and energy. The obtained formulas provide a clear identification of DMI and long-range dipolar interaction as two physical mechanisms determining skyrmion size and stability.

Figure 1

Dependences of the skyrmion characteristics on the parameters obtained from the asymptotic analysis for $\left|\kappa \right|,\delta \ll \sqrt{Q-1}$: (a) the dimensionless skyrmion radius $r_0$ from Eq. (6); (b) the skyrmion collapse energy barrier $\mathrm{\Delta }{E}_0$ from Eq. (8); and (c) the rotation angle $|{\theta }_0|$ from Eq. (5). The solid line shows the transition from Néel to mixed Néel-Bloch skyrmions governed by Eq. (9). The dashed line corresponds to the boundary of the region defined in Eq. (3) in which existence of skyrmion solutions is guaranteed. The dotted line shows the parameters at which the skyrmion radius achieves its minimum value as a function of $\delta$ according to Eq. (11).

Figure 2

Skyrmion phase diagram for $A=20$ pJ/m, $D=0.3$ mJ/${\mathrm{m}}^2$, and $d=1$ nm. The dark red zone is the domain of existence of our skyrmion solutions (see Sec. 3d for a complete description of the different zones and lines).

Figure 3

Dependences of the skyrmion characteristics in the low-DMI regime. The parameters are $A=20$ pJ/m, $M_{\mathrm{s}}={10}^5$ A/m, and $K_{\mathrm{u}}=6346$ J/${\mathrm{m}}^3$ corresponding to $Q=1.01$. (a) The skyrmion radius $r_{\mathrm{sky}}$. (b) The normalized skyrmion collapse energy barrier $\mathrm{\Delta }{E}_{\mathrm{sky}}/\left(k_{\mathrm{B}}{T}_{293\mathrm{K}}\right)$. (c) The rotation angle $|{\theta }_0|$. (d) The skyrmion profile obtained numerically for $d=5$ nm and $D=0.018$ mJ/${\mathrm{m}}^2$, corresponding to the white dots in panels (a)–(c), using MuMax3 [57] on a $4096\times 4096{\mathrm{nm}}^2$ square domain subject to periodic boundary conditions, with the mesh size of $4\times 4\times 5{\mathrm{nm}}^3$. The image in panel (d) is constructed by superimposing the in-plane magnetization ${\mathbf{m}}_{\perp }$ represented with arrows and the out-of-plane magnetization $m_{\parallel }$ represented by a color map. The lines in panels (a)–(c) are the same as in Fig. 1.

Figure 4

Dependences of the skyrmion characteristics in the intermediate DMI regime. The parameters are $A=20$ pJ/m, $M_{\mathrm{s}}={10}^5$ A/m, and $K_{\mathrm{u}}=1.26\times {10}^4$ J/${\mathrm{m}}^3$ corresponding to $Q=2$. (a) The skyrmion radius $r_{\mathrm{sky}}$. (b) The normalized skyrmion collapse energy barrier $\mathrm{\Delta }{E}_{\mathrm{sky}}/\left(k_{\mathrm{B}}{T}_{293\mathrm{K}}\right)$. The gray zone corresponds to the region where $r_{\mathrm{sky}}<d$.